Both aspects, the double zero of \(\mathcal {F}(h)\)  (see subsection REF  below, for example) and the possibility of using correlations between \(a_i\)  coefficients to distinguish SMEFT from HEFT from experimental data (see subsection REF ) have been separately discussed in the last years. We here provide an integrated discussion with full detail, putting less weight on geometric aspects and more in field-theory and particle physics ones, more familiar to the typical reader, and make several new contributions.
Finally, it is also worth commenting that similar relations exist between the coefficients of the nonderivative \(V_{\rm HEFT}\) 
potential, since a valid SMEFT description of \(V\)  must obey a series of conditions analogous to those for the flare function \(\mathcal {F}(h)\)  [1]}. Assuming SMEFT's validity would then also impose important correlations between the coefficients of the potential –trilinear, quartic, etc.– (see e.g. [2]} for a HEFT phenomenological analysis). This is mostly beyond the scope of this article and will certainly be studied in future work, but we present a slim discussion in Appendix .
